the value of electronic hyperfine anomalies to nuclear and ... - CiteSeerX

JOURNAL DE PIIYSIQUE

Colloque C4, suppltment au no 11-12, Tome 34, Novembre-Dtcembre 1973, page 69

THE VALUE OF ELECTRONIC HYPERFINE ANOMALIES TO NUCLEAR AND SOLID STATE PHYSICS N. J. STONE Mullard Cryomagnetic Laboratory, Clarendon Laboratory, Oxford, UK

ResumC. - L'anomalie hyperfine a ete principalement consideree comme un paramktre interessant en connexion avec la theorie du moment dipolaire nucleaire. I1 est montre que cette application est limitee par I'etat actuel de la theorie de cet effet. En contrepartie, la comparaison de mesures d'anomalies dans differents environnements Clectroniques conduit a des informations concernant I'origine des interactions magnetiques hyperfines et ceci sans qu'il soit necessaire, a priori, de requkrir des calculs quantitatifs precis de I'anomalie. Des resultats recents d'anomalies, pour les isotopes de Ir et Au dilues dans des alliages ferromagnetiques, indiquant la formation d'un moment local, pres de Ir et Au, sont discutes. Abstract. - The hyperfine anomaly has been primarily considered as a parameter of interest in connection with nuclear dipole moment theory. It is shown that the present state of the theory of the effect limits this application. In contrast the comparison of anomaly measurements in different electronic environments yields information concerning the origin of magnetic hyperfine interactions without requiring a precise quantitative calculation of the anomaly a priori. Recent anomaly results on dilute Ir and Au isotopes in ferromagnetic alloys, which indicate local moment formation at the Ir and Au site, are discussed.

1. Introduction. - This paper reviews, in outline, the basic theory of the hyperfine anomaly in electronic hyperfine interactions. The complexity of the theory arises from the need to consider the effects of the distribution over the nuclear volume of nuclear magnetism (DNM) and of nuclear charge, the latter determining the variation of the clectron wavefunctions within the nucleus. The limited value of the anomaly as a parameter to assist in the choice of appropriate nuclear models is illustrated briefly. It is shown that in favourable cases the anomaly can give more information concerning atomic and solid state problems. Its measurement in different environments can lead to separate knowledge of contact terms and non-contact or orbital terms in the hyperfine interaction. Since the separation is only accurate to the precision with which the ratio of hyperfine anomalies in two environments can be measured, the conditions for existence of large anomalies are reviewed. Details are given of the use of anomaly measurements in this way for Ir and Au isotopes in dilute ferromagnetic alloys, and the interpretation of the results is outlined. With the increasing frequency of precise hyperfinc interaction measurements on excited nuclear states, often giving wider variation in D N M than found in stable ground states, it is t o be expected that the use of the hyperfine anomaly in this way will become more important. 2. The hyperfine anomaly. - Magnetic hyperfine interactions can be produced either by spin or orbital dipolar interactions for electrons with wavefunctions of non-spherical symmetry, o r by the (t contact ))

interaction of electrons having spherically symmetric wavefunctions which are finite at the origin. Considering the nucleus as a point dipole, Fermi and Segre [I] derived the expression

8 7-r Contact hyperfine field = - 7p.

1 $(O) l2

where p, is the Bohr magneton and $(0) is the electron wavefunction amplitude at the origin. A classical derivation is given by Ferrell [2]. The contact interaction arises for s electrons, and for heavy atoms where relativistic components in the wavefunctions are important, for p,,, electrons also. This expression predicts that the contact hyperfine interactions Wi of two isotopes of a single element would be in the ratio of their nuclear dipole moments i.e.

This relation does not hold in practice, the deviation being characterised by a (( hyperfine anomaly )) 'A2, such that

The origins of the anomaly lie in the breakdown of the (1)

for

Those readers who feel strongly on the matter can read if they so wish. The use of

for

hyperfine field

throughout this paper avoids confusion with current usage and its accuracy will not be discussed here.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973409

N. J. STONE

C4-70

point nucleus assumption which is implicit in eq. (1). The hyperfine field H varies over the finite nuclear volume since the electron wave function $(r) is not constant, and the DNM is non-uniform. Thus the general expression for the contact interaction energy should be

w.=

-

J

~(7)

dr

nucleus

where d7 is an element of nuclear volume. Since the D N M can be different for different isotopes we express the interaction energy in the form

where W,,,,, is a fictitious point dipole interaction and ci an empirical parameter describing the deviation from it for a given isotope i.

2 . 1 THEORETICAL CALCULATION OF E . - The problem was first formulated by Bohr and Weisskopf [3]. They derived a result of fundamental importance, namely that the interactions of the contact field with spin and orbital contributions to the nuclear dipole moment are essentially different. The element of nuclear spin magnetisation at radius r, from the origin interacts with the field at that radius H(r,) to give

In contrast the nuclear orbital magnetisation acts like a current loop and its energy depends upon the mean field through the orbit H ( r < r,) to give

A non-relativistic derivation of these results has been given by Sorensen [4]. Thus since H(r,) varies as $(r) and hence falls from a maximum at the origin, the deviation from a point nucleus interaction is larger for spin contributions than orbital contributions to the D N M at the same distance from the origin. Collective orbital contributions are treated in the same way as single particle orbital terms. Contributions to E are then written

I

l2

where

and

In these expressions a, and crL are the fractional spin and orbital contributions to the nuclear dipole moment, thus a, + a, = 1. < Ks >,, and < KL >,, contain appropriate averages of powers of the electron density distribution over the nucleus. Tbe

Q term in tS allows for a non-spherical contribution to the spin interaction in aspherical nuclei and was introduced by Bohr [5]. Thus the calculation of c depends upon the nuclear moment through the breakdown into spin and orbital contributions and their distributions within the nucleus, and also upon the electron wavefunctions. Tabulations to yield < Ks >,, and < K L >,, for both s , ~ , and p l i 2 relativistic electron wavefunctions based on a uniform charge distribution are given by Eisinger and Jaccarino [6]. Stroke, Blin-Stoyle and Jaccarino [7] have shown the sensitivity of E to the assumed charge distribution to be of order 30 % by comparing calculations for uniform volume charge, uniform surface charge and diffuse o r trapezoidal charge distributions. More recent calculations of electron wavefunctions over the nuclear volume are available, the prime motivation being interpretation of the isomer shift measured by Mossbauer spectroscopy. The isomer shift is the electric monopole analogue of the magnetic hyperfine anomaly, its concern being the electrostatic interaction of nuclear and electronic charge distributions. A recent review of these improved wavefunction calculation techniques by Kalvius [8] serves only to emphasize the level of the demands such complex interactions make upon theory and how limited is its current success. As regards the nuclear models, calculations have been made using pure shell model [3], shell model with weakly coupled core contributions [5], shell model with configuration mixing [7] and Nilsson [9] wavefunctions. It was hoped that measurements of hyperfine anomalies might assist the choice of appropriate nuclear models. This hope has not materialized since experimentally the effects are generally small, and they involve both the electronic and nuclear wavefunctions in a sophisticated way. Furthermore, is a notional quantity, the only measurable since WpOinl parameter, 'A2 involves the difference of E for two nuclear states. Comparison with experiment is considered further in section 3. 2 . 2 THEMAGNITUDE OF c A N D OF ' A ~ .- Figure 1, due to Eisinger and Jaccarino [ 6 ] , shows the radial dependence of the sl12 electron density for a point nucleus and for a uniform finite charge density distribution. An important result is that, to the order (nuclear radius/Bohr orbit r a d i ~ s )the ~ sIi2 electron radial dependence is the same for all principal quantum numbers. This means that the anomaly will be independent of the shell in which s-electron interaction originates. For a given 2, pIl2 electron anomalies are smaller than sl12 and they are only significant for Z > 50. A calculation by Eisinger and Jaccarino [6] assuming uniform nuclear charge and DNM over a sphere of radius Rc gave Ave

THE VALUE O F ELECTRONIC HYPERFINE ANOMALIES T O NUCLEAR

C4-71

THE CASE OF INFINITE HYPERFINE ANOMALY POINT NUCLEUS

I

I

Hnon-contact

"PI

(non-s electron)

Elemental nuclear

Hcontact spln and o r b ~ t a l moments

I

I

c,vs(r)

(5-electron)

Spin I

orbital Varlat~onof the S-electron denslty over the nuclear volume FIG. I . - Radial dependence of the s112 electron density a) for a point nucleus and b) for a uniform charge density distribution.

where a, is the first Bohr orbit radius, R is the radial co-ordinate and the average is over the nuclear volume. Since, even for Z = 50, this gives E less than 2 % we can see why the hyperfine anomaly 'AZ = E, - 8, is seldom more than 1 % and for light nuclei generally less than 0.1 %. A recent tabulation of anomaly measurements is by Fuller and Cohen [lo]. However, there exist characteristic situations in which the anomaly can be much larger. These occur when the nuclear moment, which is always the vector sum of the spin and orbital contributions, is the small resultant of two large and opposing terms. Such cases arise in odd Z nuclei of low spin with the odd proton in (( jacknife )) j = 1 - -1orbitals. These are illustrated for the p,,, and d3/, cases in figure 2 where the unit on the vertical scale is the resultant moment in each case.

1-1-52 10.

Spin

I p(rl[Hpt-H(r;l

FIG. 3. - The case of infinite hyperfine anomaly. This schematic diagram shows a) the exact cancellation of the nuclear moment contributions and b) the different spin and orbital departures from a point nucleus hyperfine interaction leading to non-zero resultant.

The contact hyperfine interaction is given by

which is non-zero since E, # E , ~ . Such a nucleus thus has a non-zero contact hyperfine interaction despite having zero magnetic dipole moment. The hyperfine anomaly 'A2 between this isotope and any other would therefore be infinite. The infinite anomaly is of course a singular case, but it is generally true that large departures from the point nucleus interaction (large c ) arise for p,,,, d,/? and f,,, odd proton nuclei, the largest for those with h ~ g hZ. Large anomalies then occur when comparisons are made between these states (of large E,) and other isomers or isotopes of the same element having very different D N M (and thus small E,) when

al 5.

l

as L

a!

- 5'

-10.

as

ph

-

d3 2'

Spin and o r b ~ t acontrl l butlons to the magnetrc moment for p and d nucle~ k2

3/2

FIG. 2. - Cancellation of spin and orbital contributions to pl/2 and d3/2 odd proton nuclear dipole moments. The unit on the vertical scale in each case is the resultant nuclear moment.

As an extreme example, figure 3 illustrates schematically the case of exact cancellation of the nuclear moment contributions, i.e. P = k + w ~ = o .

The largest measured anomalies (of order 5-10 %) involve d,/, proton states in gold and iridium isotopes (see below). For intermediate Z elements anomalies of 2-3 o/, have been measured between isotopes of silver (odd proton p,,,). 3. Application to nuclear physics : Comparison of measured anomalies with nuclear model calculations. - A s mentioned at the end of section 2.1 various measurements of hyperfine anomalies have been made in the hope that comparison with calculations based on different model wavefunctions having different DNM would indicate a preferential choice of nuclear model. It must be emphasized that the measurement of a large A requires that the two isotopes have very different DNM, and that conversely a small A can occur between two isotopes, for both of which the

N. J. STONE

Comparison of theory and experiments for 312A112'9 3 ~for r various electron wavefunctions and nuclear models (Perlow) Comment

Model -

-

1) Bohr & Weisskopf model 2) Pure single particle model with uniform nuclear charge

25.0 %

3) Pure single particle model with trapezoidal nuclear charge

14.0 %

4) Weak coupling model with core excitation

27.0 %

5) Nilsson model

20.0 % 7.2 %

Measured value (Mossbauer)

/

1

Note elfect of nuclear charge shape on electronic wavefunctions and hence onA

Emphasizes that large A's occur where strong cancellation makes moment theory difficult

Comparison of theory and experiment for 19'A19' (Gold) on various nuclear models 197~199

Bohr and Weisskopf Weak coupling model (Bohr) Configuration mixing shell model Measured value (Atomic Beam) deviation ei from a point nucleus interaction may be large, if their DNM are closely similar. Experimentally a reliable measurement of A for comparison with theory requires that the origin of the hyperfine interaction measured be well understood so that division into contact Wc and non-contact or orbital Wncparts can be made. This is necessary since the non-contact interaction, being uniform over the nuclear volume t o a high degree of accuracy, will not contribute to A and must be subtracted, thus in general and

supercedes eq. (2). Furthermore, if both slJ2 and p,,, terms contribute to the contact interaction their relative contributionsf, and f, must be known, giving 'A:

=

f , 'A2(s)

+ fp 'A2(p).

As examples of the limited value of this procedure we consider briefly the cases of Ir and Au isotopes. Both show large anomalies which can be accurately measured. 3.1 Ir ISOTOPES. -The anomaly between the ground states of the stable isotopes lg1Ir and '931r

Comment

-

-

6.5 % 4.0 % 4.9-6.6 "/,

All models give correct order of A, Measurement is not model selective

deduced from NMR measurements in Ir metal [Ill, [12] and paramagnetic resonance is - 0.2 (2) %. Since both are odd proton d3/, this small value confirms they that have closely sim~larDNM. Of much greater interest is the Mossbauer effect measurement of A 4 excited state of between the 3' ground state and ' lg31r by Perlow et al. [13]. The crucial hyperfine interaction experiments were done in IrF, and the magnetic moment ratio was obtained from measurements in an external magnetic field applied to nonmagnetic iridium metal. After correction for the noncontact interaction in IrF, the value of 3J2A1/2was 5.8 (6)%. Perlow [9] has reported calculations based on various nuclear models and electron wave-functions. These are summarised in table I. 3.2 Au ISOTOPES. - Atomic beam studies of Au yield useful A values directly since the electronic hyperfine interaction is due entirely to the outer 6s electron contact term. Measurements by Van den 99~~ Bout et al. [14], [15] on 1 9 6 ~ 1 9 7 ~ 1 9 8 ~ 1yielded large values of A. The value for l g 7 ~ l gi9s compared with various model calculations in table I1 and calculations by Bacon [I61 using the Eisinger and Jaccarino theory for this and other pairs of Au isotopes are given in table 111. The latter shows clearly that the large anomalies arise from large ei (-- 10 %) associated with lg7Au and lg9Au, both single proton

THE VALUE OF ELECTRONIC HYPERFINE ANOMALIES TO NUCLEAR

A

(2-73

Calculated and measured hyperfine anomalies for different Au isotopes

in any other environment 'A2(B) can be used to separate the contact and non-contact contributions to H!;('B) through the relations

Cal. e(%) E&J

and

197AA

19 SAA

Exp.

197 10.3 198 0.4 196 0.4 196m -1.9 198111 -1.9 200111 - 1.9 199 4.3

..-

+ 8.53 (8) + 8.72 (24) (8)

Cal. Exp. - 9.0 - 7.96 (8) (a) --

(a)

+3.7(2)(Q)

0.0

+ 2.3 + 2.3 + 2.3 -3.8

-

+ 0.2 (3) (b) -4.5(3)(5)

d,/, nuclei, contrasting with smaller ti ( N 1 %) for the odd-odd isotopes and isomers with less cancellation in their DNM. 3 . 3 CONCLUSION. - For both the Au and Ir systems, while it is clear that calculations of E , can give qualitative agreement with experiment no strong indication of nuclear model results, not least because of the sensitivity of the calculation to the assumed electronic wavefunctions. This conclusion holds for other systems as discussed in previous reviews [6], [7]. More detailed calculations could be made, however, as Perlow has remarked [9], it is for just those cases of strong cancellation, which test the accuracy of theories of nuclear magnetism beyond their present level, that the larger and more precisely known anomalies arise.

4. The application of hyperfine anomaly measurements in solid state physics. - Although it is clearly difficult to make quantitative use of measured A's to assist in questions of nuclear models, the hyperfine anomaly can be of value in elucidating the problem of the origin of a particular hyperfine interaction. This arises since only contact terms due to slI2 or relativistic pIl2 electrons contribute to A. Provided some system, A, exists in which the proportion of a measured hyperfine interaction due to a single contact term and any combination of non-contact terms is known it follows that the anomaly characteristic of that contact hyperfine interaction A, can be calculated. For simplicity we will proceed on the assumption that plI2 terms may be neglected unless specified. If the interaction for isotope 2 be written

then one can show that the relation

(4lAZ A2 - fC2) A - H g (A)

1

holds between the measured anomaly 'A:, the effective field H~;'(A) as defined above, the contact term contribution H,(,)(A) and the pure s electron contact anomaly 1 2 A,. Once 'A: is known, then a measured anomaly

'A: H~;;(B)

=

'A; H:"

Note that H,, (B) will be the same for all isotopes, but that both H, and Herrwill be isotope dependent. In contrast to the use of A in nuclear physics, here no a priori derivation of 'A: is required. Rather it is used simply as an indicator and only ratios of measured anomalies in different environments are involved in the results. It is important also to re-emphasize the independence of 'A: of the s electron shell involved, so that systems involving outer s electron interactions or contact interactions produced by inner core s electron polarization can be dealt with alike. There has been little exploitation of this possible application of hyperfine anomaly measurements since measured A ratios are frequently too inaccurate to be of value, although the capability has been recognised for example in atomic beay [17] and ENDOR [18] studies. Apart from accuracy the fact that stable isotopes of an element often have similar DNM and the lack of known systems ( A above) t o cc calibrate D the measured A's also inhibit the prospect of general use of A in this way. Recent increases in accuracy of measurement of hyperfine interactions of excited states and radioactive isotopes by sensitive Mossbauer experiments and by nuclear magnetic resonance detected by radioactive methods have widened the range of nuclear states for which A measurements are possible and have led to a first concerted use of the anomaly in solid state physics. This has been in the field of hyperfine interactions in ferromagnetic materials for the elements Ir and Au for which, as described above, the anomalies are large and the value of 'A: can be established. Related work on Eu isotopes has been done by Crecelius and Hufner [19] and by Aiga et al. [29] on Cu, Sb and Ir isotopes. 4.1 DILUTEALLOYS OF Ir IN Fe, Co, A N D Ni (MOSSEFFECT). - Since Mossbauer spectra yield values of the hyperfine interaction for both nuclear states involved in the gamma transition under study a single measurement includes a value for gAex between these states provided their moments are known. Thus Perlow et al. [13], having established the value of 3 / 2 ~ 6 / 2for the 1931rtransition, were able to interpret a spectrum published by Wagner et al. [20] on IrFe to yield Hc and H,,. More recent results due to Wagner and Potzel [21] are given in table IV, where the values of Hcf,and H, are those for the spin f excited state. BAUER

4.2 DILUTE ALLOYS 01: AU I N Fe, Co ( N MR/ON). As a result of atomic beam studies values of the s electron contact anomalies between 196*,"7.1983'99A~

N. J. STONE

resonance shift is caused by an electric quadrupole interaction [24]. Results on lg6Au in Fe [25], which closely agree with those on 19'Au, are also given in table V.

TABLEIV

Interpretation of hyperfine fields at iridium in ferromagnetic metals using hyperfine anomaly data [21]

4.2.1 Interpretation of the results in ferromagnetic alloys. - The analysis of hyperfine anomaly measurements in ferromagnetic alloys of Ir and Au has experimentally confirmed that the hyperfine interactions are largely produced by contact interactions. This result was fully expected, being due to exchange polariContact anomaly 3 1 2 ~ 6 ' ~ -k 6.5 (6) %. sation of core s electrons of the impurity atoms by interaction with the polarised conduction electrons of the host. The magnitude and sign of the non-contact are known (section 3.2). Figure 4 shows NMR spectra term is of more interest. Gehring and Williams [26] of 1 9 8 A and ~ 199Au in dilute Fe alloys detected by have analysed the results on both systems. They have the change in gamma ray distribution from nuclei shown first that the results cannot be explained in polarised at 0.01 K [22]. The '99Au resonance freterms of a p,,, contribution to the contact interaction quency predicted on the basis of the atomic beam of reasonable magnitude. Secondly they find the anomaly value, 198A199= -- 4.5 %, is marked by explanation in terms of an orbital dipolar field of a vertical arrow. The discrepancy of 1.7 MHz is order + 200 kOe to be consistent both in sign and clearly seen and the current best value of the 198A199 magnitude with the unquenched 5d orbital moment in this system is - 5.3 (3) %. These data and similar to be expected for these heavy impurity atoms which results in cubic C o yield the values of H,,,, etc given have strong spin-orbit coupling. Further evidence in table V. Johnston [23], by further careful experifor this explanation comes from the observation of mentation, has eliminated the possibility that the associated weak electric quadrupole interactions in the Ir alloys [21], [27], [28] which have been shown to be isotropic by an NMR/ON study of a single crystal IrNi- sample [28]. 7

l;!.ut/or; -o f i'''n,J

and

'"A!.

1"

F